/sci/ - Science & Math

[Anonymous No.6622635]

In the algebraic approach to p-adic numbers

http://en.wikipedia.org/wiki/P-adic_number#Algebraic_approach

how does the strange metric arise?
I see no mention of it there, so why wold you think about introducing it at all?
Is it by first constructing an order in this field and somehow reading it off?

[Anonymous No.6622664]

There is not metric in this construction; only the field structure.
You could then give it the discrete metric if you wanted.

[Anonymous No.6622679]

>>6622664
Okay.
I thought the metric is central to p-adic number systems.
If not, or actually in general, why is it interesting? Because it has a universal property in the theory of fields?

[Anonymous No.6624132]

I heard there is p-adic Lie groups.

What are the features of those things?
Is there anything "as big" or smooth as SU(2), for example. Or are those things essentially degenerate (e.g. like finite fields, which I see as some collapsed stucture)?

[chroma No.6624271]

>how does the strange metric arise?

modular arithmetic. an integer's residues mod p, p^2, p^3, p^4, ... are increasingly more narrow specifications of that particular integer. occasionally (with, say, binomial sums, or roots of polynomials) one has a congruence mod a power of p, and wants to know if or when we can extend it to higher powers of p (a stronger level of specification). on this view, the higher the power of p dividing x-y, the "closer" x and y are to each other, as far as the prime p is concerned.

since "the highest power of p in x's factorization" is a map satisfying v(xy)=v(x)+v(y), setting N(x):=a^v(x) for some base a gives a multiplicative map. by letting a be in the interval (0,1), this map is also an absolute value (it satisfies the triangle inequality), which induces a metric via d(x,y):=|x-y| that measures how far integers are from each other "according to p."

this metric applies just as easily to rational numbers. completing Q with respect to the Euclidean metric gives R (take the ring of Cauchy sequences of rationals with coordinatewise operations and quotient by the maximal ideal of sequences converging to zero). completing Q with respect to the p-adic metric gives the p-adic numbers Q_p, and completing Z with respect to the p-adic metric gives Z_p.

[chroma No.6624275]

Note that it is not possible to make Q_p or Z_p into an ordered ring (meaning, a ring with a total order such that x<y implies x+z<y+z and a,b>0 implies ab>0, which is necessarily characteristic zero), since ordered rings cannot have square roots of negative integers whereas p-adic integers have infinitely many such square roots for any p.

>There is not metric in this construction; only the field structure.
it's certainly possible to construct Q_p as the fraction field of the inverse limit of the rings Z/p^n, and then the metric comes later, but as I described above it's also perfectly possible to construct Q_p *with* the metric.

>I heard there is p-adic Lie groups. What are the features of those things?

I am not really familiar with them. Real and complex lie groups are smooth manifolds, like the surface of a marble, but p-adic groups are totally disconnected spaces (essentially like rooted p-ary trees), which are at different extreme ends of the spectrum to any topologist. I think the interest in p-adic Lie groups is that they say things about automorphic forms and Galois representations.

[chroma No.6624289]

forgot to mention why we choose a=p^-1 when we define the p-adic absolute value. certainly any other choice of a in the interval (0,1) will induce an equivalent absolute value, meaning the topology induced from the metric it induces is the exact same. with this choice, we have the product formula <span class="math">\prod_{p\le\infty}|x|_p=1[/spoiler] (man, it's been awhile since I've used mathjax on 4chan), where |x|_inf is the usual Euclidean absolute value. this is nice when we're trying to prove, say, approximation theorems for the adeles (which is essentially a very sophisticated generalization of the chinese remainder theorem for integers, although CRT generalizes more abstractly in a different direction to other rings, even noncommutative). we also have that |x|_p is the Haar measure of the principal ideal (x) within Z_p.

fun fact: you can complete global fields (number fields - finite extensions of Q - and function fields - finite extensions of F_q(T)) with respect to absolute values induced from prime ideals in the same exact manner.

some other keywords and phrases you may be keen to search: adele ring, inverse limit, profinite group, ultrametric archimedean tree topology, local rings, discrete valuation rings, local fields

I'd love to answer any other questions anybody has, if I can

[Anonymous No.6624293]

>tfw almost finished my BSc majoring in pure maths and still can't into p-adic analysis

[Anonymous No.6624358]

>>6624271
>>6624275
>>6624289
Thanks for the elaborations.
I don't have a perfect picture about why the p-adic norm works, but from what you wrote I have a feeling how it's related to modular arithmetic and how it measures distances according to some prime - it encodes information about some "reduced numbers", after information has been thrown away (compared to ordinary arithmetic and a norm being an ordinal value, visualizable as the distance on a straight line, always away from the zero point).

I never really had to learn about p-adic numbers - they seem to be amongst the "most useless" concepts that still got researched a lot. (Because, as you say, Lie-theory is usually an inherently smooth busyness, I was surprised to learn there even is such a thing as p-adic Lie theory.)
I'm interested in their relationship to the general theory of ring. I know that

I read in a review that Fraenkel worked on p-adics
http://en.wikipedia.org/wiki/Abraham_Fraenkel#Mathematician
I encountered them a second time here
http://math.stackexchange.com/questions/38517/in-relatively-simple-words-what-is-an-inverse-limit/38522#38522
And I generally want to come at it form the algebraic side and like to get an intuition for why they are there. The norm doesn't play a role there and that's why I asked.

[Anonymous No.6626250]

bmp